A random sampling technique is employed to obtain the ground-state configuration energy of an overcharged macroion in that the magnitude of the total counterion charge exceeds the magnitude of the macroion charge. The technique is based on minimizing the total electrostatic energy of the macroion and counterions in salt-free conditions. The algorithm easily reproduces the ground-state energies of an overcharged isolated spherical macroion obtained from Molecular Dynamics simulation in earlier published literature (Messina, R., Holm, C.; Kremer, K Phys. Rev. E 2001, 64, 021405) for low to intermediate values of the macroion charge, while at higher macroion charge the calculated energies lie below those of the earlier results. The calculations are extended to nonspherical macroion geometries such as spherocylinders and spheroids, and substantial effects on the energetics due to the geometries are noted. A detailed study of counterion distributions is also done for macroions of different charges and geometries.